Why it happens: "Subscription Update" is Stripe's own wording, not a JourneyCARE setting. The subscription is created at the first purchase, and every later automatic rebill acts on that existing subscription, so Stripe records it as an update. The label cannot be changed. The Line Item details are in JourneyCARE, even if they do not show in your Stripe account.
How to see the membership type: export your transactions as a CSV from Payments > Transactions using the Download icon. The export includes line-item columns, and Line item name is the product name behind each charge. See Export or download transactions as CSV for the full column list.